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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b4">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W</font> &amp; <font color="Olive" title="b1">P</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp;  not <font color="Olive" title="b4">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Olive" title="b2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Olive" title="b1">P</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>)</span>)</span> &amp; <font color="Olive" title="b4">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Q</font> holds <br/><font color="Olive" title="b2">P1</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Olive" title="b3">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">P1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> &amp; <font color="Maroon" title="c2">P</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp;  not <font color="Olive" title="b2">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c2">P</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>)</span>)</span> &amp; <font color="Olive" title="b2">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Maroon" title="c3">P1</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Olive" title="b1">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c2">P</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp;  not <font color="Olive" title="b1">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c2">P</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>)</span>)</span> &amp; <font color="Olive" title="b1">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Q</font> holds <br/><font color="Maroon" title="c3">P1</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c2">P</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp;  not <font color="Maroon" title="c5">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c2">P</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c5">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Q</font> implies <font color="Maroon" title="c3">P1</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">Q</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font>
 <span class="kw">and </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">and </span><br/><a NAME="E3:69"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 not <font color="Maroon" title="c5">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E4:69"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c3">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c2">P</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:69"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">P1</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">Q</font></span><br/>

<a NAME="E6:69"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="connsp_1.html#D3" onmouseover="rs('connsp_1/D3')" onmouseout="rh()">CONNSP_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A7</font></span>:  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c2">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>) = 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T13" onmouseover="rs('subset_1/T13')" onmouseout="rh()">SUBSET_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="xboole_1.html#T83" onmouseover="rs('xboole_1/T83')" onmouseout="rh()">XBOOLE_1:83</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">P0</font> = <font color="Maroon" title="c2">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">W0</font> =  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c6">P0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="topreal3.html#T8" onmouseover="rs('topreal3/T8')" onmouseout="rh()">TOPREAL3:8</a></span>;<br/>
<a NAME="E10:69"/>
<font color="Maroon" title="c6">P0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="subset_1.html#T23" onmouseover="rs('subset_1/T23')" onmouseout="rh()">SUBSET_1:23</a></span>;<br/>
<a NAME="E11:69"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a></span>;<br/>
<a NAME="E12:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<font color="Maroon" title="c6">P0</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="connsp_1.html#D3" onmouseover="rs('connsp_1/D3')" onmouseout="rh()">CONNSP_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E55">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font> &amp;  not <font color="Olive" title="b1">D</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:69_2"/>
 for <font color="Olive" title="b1">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P1</font> holds <br/><font color="Olive" title="b1">D</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:69_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">W0</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E53">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:69_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="connsp_3.html#T1" onmouseover="rs('connsp_3/T1')" onmouseout="rh()">CONNSP_3:1</a>, <a class="ref" href="tbsp_1.html#T14" onmouseover="rs('tbsp_1/T14')" onmouseout="rh()">TBSP_1:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:69"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/>
<a NAME="E15:69"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E51">A5</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:69_3"/>
<font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:69_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:69_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E14:69"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E15:69"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">Q1</font> = <font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E18:69"/>
 not <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c8">Q1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
;<br/>
<a NAME="E19:69"/><span class="kw">then </span>
 <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c6">P0</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E14:69"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E12:69"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="connsp_3.html#T9" onmouseover="rs('connsp_3/T9')" onmouseout="rh()">CONNSP_3:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:69"/>
<font color="Maroon" title="c3">P1</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E13:69"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="jordan2c.html#T14" target="_self" onmouseover="rs('jordan2c/T14')" onmouseout="rh()">Th8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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